# Alternating group:A5

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This particular group is the smallest (in terms of order): simple non-Abelian group

## Definition

The alternating group $A_5$ is defined in the following ways:

• It is the group of even permutations (viz, the alternating group) on five elements
• It is the von Dyck group (sometimes termed triangle group) with parameters $(5,3,2)$
• It is the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron)
• It is the projective special linear group of order two over the field of five elements, viz $PSL(2,5)$

## Bigger groups

### Groups having it as a subgroup

The alternating group is a subgroup of index two inside the symmetric group on five elements. It is also of index two in the full icosahedral symmetry group, which turns out not to be $S_5$, but instead the direct product of $A_5$ and the cyclic group of order two.

### Groups having it as a quotient

The alternating group is a quotient of $SL(2,5)$ by its center. Hence, it is the inner automorphism group of $SL(2,5)$. $SL(2,5)$ is also the universal central extension of the alternating group.